1st Test - Reasoning And Patterns

DEDUCTIVE vs. INDUCTIVE REASONING

Huh?

Deductive Reasoning

• Deductive Reasoning – A type of logic in which one goes from a general statement to a specific instance.

• The classic exampleAll men are mortal. (major premise)

Ben is a man. (minor premise)

Therefore, Ben is mortal. (conclusion)

Deductive Reasoning

Examples:1. All students eat pizza.

Claire is a student at ASU.Therefore, Claire eats pizza.

2. All athletes work out in the gym. Barry Bonds is an athlete. Therefore, Barry Bonds works out in the gym.

Deductive Reasoning

3. All math teachers are over 7 feet tall.

Mr. D. is a math teacher.

Therefore, Mr. D is over 7 feet tall.• The argument is valid, but is certainly not true.• The above examples are of the form

If p, then q. (major premise)

x is p. (minor premise)

Therefore, x is q. (conclusion)

Venn Diagrams

• Venn Diagram: A diagram consisting of various overlapping figures contained in a rectangle called the universe.

U

This is an example of all A are B. (If A, then B.)

B

A

Venn Diagrams

This is an example of No A are B.

U

AB

Venn Diagrams

This is an example of some A are B. (At least one A is B.)

The yellow oval is A, the blue oval is B.

Example

• Construct a Venn Diagram to determine the validity of the given argument.

#14 All smiling cats talk. The Cheshire Cat smiles.

Therefore, the Cheshire Cat talks.

VALID OR INVALID???

ExampleValid argument; x is Cheshire Cat

Things that talk

Smiling cats

x

Examples

• #6 No one who can afford health insurance is unemployed.

All politicians can afford health

insurance.

Therefore, no politician is unemployed.

VALID OR INVALID?????

Examples

X=politician. The argument is valid.

People who can affordHealth Care.

Politicians

X

Unemployed

Example

• #16 Some professors wear glasses. Mr. Einstein wears glasses.

Therefore, Mr. Einstein is a professor. Let the yellow oval be professors, and the blue oval be glass

wearers. Then x (Mr. Einstein) is in the blue oval, but not in the overlapping region. The argument is invalid.

Inductive Reasoning

Inductive Reasoning, involves going from a series of specific cases to a general statement. The conclusion in an inductive argument is never guaranteed.

Example: What is the next number in the sequence 6, 13, 20, 27,…

There is more than one correct answer.

Inductive Reasoning

• Here’s the sequence again 6, 13, 20, 27,…• Look at the difference of each term.• 13 – 6 = 7, 20 – 13 = 7, 27 – 20 = 7• Thus the next term is 34, because 34 – 27 = 7.• However what if the sequence represents the

dates. Then the next number could be 3 (31 days in a month).

• The next number could be 4 (30 day month)• Or it could be 5 (29 day month – Feb. Leap year)• Or even 6 (28 day month – Feb.)

Functional Relationships

I know this!

180

180

180

180 180

180

? ?

# of Sides Sum of Interior Angles

3 180

4 360

5 540

6

7

…

n

1. Is the following relation a function?

2. What are the independent and dependent variables?

3. What is the constant rate of change you see in the table?

5. What is the formula for the sum of the interior angles of a polygon with n sides?

6. Certain values of n don’t make sense for this formula. What is the domain for this formula?

7. Restricting yourself to values in this domain, certain values are expected. What is the range for this formula?

8. What is the sum of the interior angles of a 18-sided figure?

9. How many sides does a polygon have if the sum of the interior angles is 2340?

10. Is this an example of inductive or deductive reasoning?